using angle MOX=135, construct an angle of 67.5 with the help of compass and ruler.

It's way to easy as you just need to bisect it.....
Use the following method-

Place the point of the compass on O, and swing an arc AB. Then, with B as center and AB as radius, draw
an arc. Keeping the same radius and with A as center, draw an arc that will intersect the first; call that point
of intersection C, and draw OC.

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